{
  "annotation_id": "e7564cad-4bf3-4bde-8d66-d378dec9da1d",
  "date_created": "2026-02-17T05:53:08.612000Z",
  "date_modified": "2026-02-17T05:53:08.612000Z",
  "file_hash": "1eea55f35dfc43916a40117af5e344c20067f3e1ae87beacd2c8725308586562",
  "private": false,
  "record": {
    "abstract": "This paper presents the study of the symplectic hulls over a non-unital ring $ E= \\langle \\kappa,\\tau \\mid 2 \\kappa =2 \\tau=0,~ \\kappa^2=\\kappa,~ \\tau^2=\\tau,~ \\kappa \\tau=\\kappa,~ \\tau \\kappa=\\tau \\rangle$. We first identify the residue and torsion codes of the left, right, and two-sided symplectic hulls, and characterize the generator matrix of the two-sided symplectic hull of a free $E$-linear code. Then, we explore the symplectic hull of the sum of two free $E$-linear codes. Subsequently, we provide two build-up techniques that extend a free $E$-linear code of smaller length and symplectic hull-rank to one of larger length and symplectic hull-rank. Further, for free $E$-linear codes, we discuss the permutation equivalence and investigate the symplectic hull-variation problem. An application of this study is given by classifying the free $E$-linear optimal codes for smaller lengths.",
    "arxiv_id": "2601.06609",
    "authors": [
      "Anup Kushwaha",
      "Om Prakash"
    ],
    "categories": [
      "cs.IT",
      "math.IT"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "title": "Symplectic Hulls over a Non-Unital Ring",
    "url": "https://arxiv.org/abs/2601.06609",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "8671a8e1-f01d-45f5-9787-9e585a51f44e",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}